File location naming hierarchy

ABSTRACT

Embodiments of methods, apparatuses, devices and/or systems for a file location naming hierarchy are disclosed.

BACKGROUND

This disclosure is related techniques for naming file locations, such asfor a file location naming hierarchy.

In a variety of fields, data or a set of data, may be represented in ahierarchical fashion. This form of representation may, for example,convey information, such as particular relationships or patterns betweenparticular pieces of data or groups of data and the like. However,manipulating and/or even recognizing specific data representations orpatterns is not straight-forward, particularly where the data isarranged in a complex hierarchy. Without loss of generality, one examplemay include a database, such as a relationship database. Typically, arelational database includes files arranging in a file naming hierarchythat provides location information so that the file may be accessed.Techniques for manipulating files, defining relationships among filesand/or moving between files in such a hierarchy continue to bedesirable.

BRIEF DESCRIPTION OF THE DRAWINGS

Subject matter is particularly pointed out and distinctly claimed in theconcluding portion of the specification. The claimed subject matter,however, both as to organization and method of operation, together withobjects, features, and advantages thereof, may best be understood byreference of the following detailed description when read with theaccompanying drawings in which:

FIG. 1 is a tree diagram illustrating an embodiment of a file locationnaming hierarchy;

FIG. 2 is a table illustrating an embodiment of names and relationshipsfor an embodiment of a file location naming hierarchy;

FIG. 3 is a table illustrating another embodiment of names andrelationships for an embodiment of a file location naming hierarchy;

FIG. 4 is a schematic diagram illustrating another embodiment of a filelocation naming hierarchy;

FIG. 5 is a schematic diagram illustrating yet another embodiment of afile location naming hierarchy.

DETAILED DESCRIPTION

In the following detailed description, numerous specific details are setforth to provide a thorough understanding of the claimed subject matter.However, it will be understood by those skilled in the art that theclaimed subject matter may be practiced without these specific details.In other instances, well-known methods, procedures, components and/orcircuits have not been described in detail so as not to obscure theclaimed subject matter.

Some portions of the detailed description which follow are presented interms of algorithms and/or symbolic representations of operations ondata bits or binary digital signals stored within a computing systemmemory, such as a computer memory. These algorithmic descriptions and/orrepresentations are the techniques used by those of ordinary skill inthe data processing arts to convey the substance of their work to othersskilled in the art. An algorithm is here, and generally, considered tobe a self-consistent sequence of operations and/or similar processingleading to a desired result. The operations and/or processing involvephysical manipulations of physical quantities. Typically, although notnecessarily, these quantities may take the form of electrical and/ormagnetic signals capable of being stored, transferred, combined,compared and/or otherwise manipulated. It has proven convenient, attimes, principally for reasons of common usage, to refer to thesesignals as bits, data, values, elements, symbols, characters, terms,numbers, numerals and/or the like. It should be understood, however,that all of these and similar terms are to be associated with theappropriate physical quantities and are merely convenient labels. Unlessspecifically stated otherwise, as apparent from the followingdiscussion, it is appreciated that throughout this specificationdiscussions utilizing terms such as “processing”, “computing”,“calculating”, “determining” and/or the like refer to the actions and/orprocesses of a computing platform, such as a computer or a similarelectronic computing device, that manipulates and/or transforms datarepresented as physical electronic and/or magnetic quantities and/orother physical quantities within the computing platform's memories,registers, and/or other information storage, transmission, and/ordisplay devices.

In a variety of fields, data or a set of data, may be represented in ahierarchical fashion. This form of representation, may, for example,convey information, such as particular relationships or patterns betweenparticular pieces of data or groups of data and the like. However,manipulating and/or even recognizing specific data representations orpatterns is not straight-forward, particularly where the data isarranged in a complex hierarchy. Without loss of generality, one examplemay include a database, such as a relationship database. Typically, arelational database includes files arranging in a file naming hierarchythat provides location information so that the file may be accessed.Techniques for manipulating files, defining relationships among filesand/or moving between files in such a hierarchy continue to bedesirable. In this context, the term file refers to any storedcollection of data that is segregated or partitioned in some manner,such as, for example, audio data, video data, computer code, text, etc.

In this context, a location naming hierarchy refers to a technique ofmethod of naming a location in which data is stored or to be stored,including, but not necessarily limited to including, a collection ofoperations to be performed upon those named locations that, if performedupon the location names, performs operations on the data stored in thosenamed locations to produce another set of data that may be stored inanother named location. One example of a location naming hierarchycommonly used is the “Unix” file hierarchy. As is well-known, in theUnix hierarchy, a particular file may have a name for an associatedlocation that comprises a collection of finite character sequences, suchas, for example, the file name “/A/B/C/***/Z”, in which, for thisexample, A,B, C, etc. comprise file names in the form of finitesequences. For the purposes of simplification of discussion, it isappropriate to think of a sequence of characters as a finite sequence ofnatural numerals. Thus, for example, a technique, such as a look uptable, may be employed to convert between finite sequences of naturalnumerals and finite sequences of characters. This may be employed, forexample, in a computer or computing system in which the computer orcomputing system performs the conversion. Likewise, in addition, thenatural numerals may likewise be converted to binary numerals so thatthey may be operated upon by the computer or computing system, althoughthis level of detail is well-understood and need not be discussedfurther. Likewise, for the purposes of this discussion, the character“/” may be replaced by a comma and, therefore, the file name above, forexample, may be represented as the following character string<A,B,C,***Z,>. These changes in nomenclature are primarily for purposesof convenience and to simplify the discussion. One of ordinary skilledin the art will appreciate that there is no loss of generality by makingthese substitutions.

With this understanding, reference is now made to FIG. 1. FIG. 1 depictsa tree intended to convey, in graphical form, a potential embodiment ofa location naming hierarchy (hereinafter referred to as “hierarchy”),such as one that may be employed in a Unix-based system. In thiscontext, the term “Unix” or “Unix-based system” generally refers to acomputing system or a computer system in which the Unix operating systemor a version of the Unix operating system is employed. Of course, it islikewise noted that the file structure described above is also common inother operating systems other than a Unix based system, such as whereother characters separate finite sequences, etc.

Referring now to FIG. 1, as illustrated, the tree includes a root node,designated 110 in FIG. 1. For the purposes of this discussion, the rootnode is referred to as being at “level 0”. Thus, here, level 0 has asingle node, the root node. In FIG. 1, directly emanating from the rootnode are four nodes on a level referred to in this context as “level 1.”For the purposes of designation in a Unix-based operating system, fromthe prior discussion, the nodes on this level may be designatedrespectively as 1, 2, 3, and 4. Thus, in FIG. 1, node 1 has referencenumeral 120, node 2 has reference 130, node three has reference numeral140, and node 4 has reference numeral 150. Continuing with thisapproach, for this example, there is a level 2, a level 3, and a level4. The level 2 includes five nodes, which may be referred to in aUnix-based operating system as 1, 2, 3, 4, and 5, respectively. Thelevel 3 includes four nodes referred to here as 1, 2, 3, and 4,respectively, and the level 4 includes five nodes referred to here as 1,2, 3, 4, and 5, respectively.

As illustrated in FIG. 1, therefore, a file represented in the tree as anode, using the nomenclature previously described, may be stored in alocation named <2, 4, 3 1>, as shown in FIG. 1 with reference numeral160. Likewise, the node that is, in this example, the file with thereference numeral 170 has a standard Unix-based operating systemdesignation or location name of <4, 1, 1>. Thus, as is clear, from theprior discussion, each of these nodes, in this example, represents aparticular file having designations or named locations in memory typicalof the Unix-based operating system. Tree 100 in FIG. 1, therefore, is agraphical illustration of the file hierarchy for this particularembodiment.

Although the scope of the claimed subject matter is not limited in thefollowing respect, a significant portion of data that is typicallystored in connection with a computer system or a computing system isstored on a device or on devices conventionally referred to as a “harddrive” or on some other storage medium in which the data is typicallyaccessed in a serial fashion. This is in contrast to memoryconventionally referred to as Random Access Memory or RAM, in which thedata may typically be accessed in a non-serial fashion. Thus, for amemory system in which data is accessed serially, such as a hard drive,as an example, accessing files is typically more time consuming thanaccessing data using RAM or Random Access Memory. A hard drive, forexample, typically locates the position of a desired file on the disk ofthe hard drive to access the particular file. Thus, accessing a fileinvolves finding the location of the particular file and then aligningthe file location with a device that is capable of reading the storeddata from that location. In comparison, for accessing data stored inRAM, the locations are, in essence, substantially simultaneously orequivalently accessible. Thus, accessing data from RAM typically takesless time at least in part because it does not involve locating the fileand aligning that location with a device capable of reading the storeddata. All or most locations in RAM are typically capable of being read.Thus, a file may be accessed once it is known which one is desired.

An aspect of knowing the location of a particular named file on a disk,for example, of a computing system employing a Unix-based operatingsystem, involves knowing the positional relationship of a particularfile to other files stored on the disk. Thus, for example, if aparticular program is executing or operating in which, initially, thefile named or designated in FIG. 1 as <2, 4, 3, 1> is accessed and, insuccession, the file named or designated as <4, 1, 1> is accessed, theoperating system, in essence, traces file locations based at least inpart on the naming convention employed, in this example by moving up theparticular file hierarchy in memory, here along a branch from the bottomnode, designated by reference 160 in FIG. 1, back to the root nodedesignated 110 in FIG. 1, and then along another branch from root node110 to node 170. In other words, in a Unix-based operating system, the“linear” relationship of the names of the particular files at each ofthe levels of the hierarchy or structure provides a mechanism to movebetween files in memory based at least in part on the particularhierarchy. This particular approach to finding the location of namedfiles to be accessed serially may be time consuming. Of course, in thisparticular context, the frame of reference for the term “time consuming”may change as technology advances. For example, in one frame ofreference, time consuming may refer a relatively short time incomparison to seconds, but a relatively long timing in comparison to thespeed of, for example, a state-of-the-art RAM. However, again, suchframes of reference may be subject to change.

It is likewise noted that an alternative approach to the one justdescribed may include maintaining the positional relationship betweenevery file to every other file. A disadvantage, however, to thisapproach is that as the number of files increase, maintaining and usingthis amount of information effectively may become intractable.

An embodiment of a method of arranging files in file hierarchy will nowbe discussed in which the relationship among files is such that movingbetween files in the hierarchy is typically less time consuming than theapproach just discussed above in connection with a Unix-based operatingsystem. In this particular embodiment, the file structure or hierarchyis based, at least in part, on a location naming convention described inthis context as employing finite multi-sets of natural numerals. Incomparison, as alluded to by previously, the file structure or hierarchyemployed in a Unix-based operating system is based, at least in part, ona location naming convention described in this context as employingfinite sequences of natural numerals. As shall be explained in moredetail below, in this particular context, a finite sequence refers to acollection of a finite number of elements indexed by the naturalnumbers. Likewise, in this particular context, a finite multi-set refersto an unordered collection of a finite number of elements. It is notedthat the term collection includes situations in which elements arerepeated. Thus, whereas for a finite sequence, the order or indexing ofthe arrangement of elements, for this embodiment, here, naturalnumerals, is an aspect of the naming convention, as well as themultiplicity or repetition of particular elements; in contrast, for afinite multi-set, the multiplicity or repetition is an aspect of thenaming convention, but the order or indexing of the elements is not.

For this particular embodiment, FIG. 2 shows a table containing fourcolumns; however, the ellipse in FIG. 2 is intended here to indicate atable of an unspecified number of total columns. As is apparent fromFIG. 2, the first column contains the natural numerals in an ascendingorder. Likewise, as shall be explained in more detail hereinafter, thesecond column contains the finite multi-sets of the natural numerals,each finite multi-set set corresponding to a natural numeral for thisparticular embodiment. More specifically, using, for this embodiment,particular rules or principles of construction, the second column has,as its elements, all possible finite multi-sets created based on thenatural numerals. These finite multi-sets corresponding to naturalnumerals, here, are referred to in this context as finite multi-sets orFMS, for convenience.

To construct the second column, for this particular embodiment, aspecific technique is employed to construct the multi-set of the naturalnumerals. Of course, the claimed subject matter is not limited in scopeto this particular embodiment and, in particular, is not limited inscope to this particular technique or these rules of construction. Thus,other rules or other approaches to constructing or creating a filehierarchy naming convention using a multi-set of the natural numerals,for example, may be employed and still remain within the scope of theclaimed subject matter. Nonetheless, as suggested above, for thisparticular embodiment, a particular finite multi-set is associated witheach natural numeral. As will be demonstrated, by repetitively applyingthese rules of construction, multi-sets of multi-sets shall be created,as explained in more detail hereinafter.

Thus, for the natural numeral, zero, for example, the empty set { } isassociated, for this embodiment, as illustrated in column two, along therow that contains zero in the first column. Likewise, for the naturalnumeral, 1 of column one, the set {0} of column two is associated, whichmay also be referred to here as “singleton zero.” For the naturalnumeral, 2, the finite multi-sets {0,0} is associated, which may bereferred to here as “doubleton zero.” Thus, in this particularembodiment, although the claimed subject matter is not limited in scopein this respect, the natural numeral zero is associated with thesimplest multi-set. The natural number one is associated with the nextsimplest multi-set, etc.

It is noted that the natural numeral 2 is the only even non-compositenumeral for this particular embodiment. An approach or rule forconstructing a finite multi-set associated with an odd non-compositenumerals is as follows, for this embodiment. For any natural numeral k,in which k is an odd non-composite numeral, a singleton set {i} isassociated with it, in which i is the index of the non-composite numeralin the well-known Kleene enumeration. For example, the set associatedwith natural numeral 3 is {1}. The set associated with natural numeral 5is {2}. The set associated with natural numeral 7 is {3}, etc.

The remaining rules refer to forming a finite mult-set associated with anatural numeral from the prior finite mult-sets associated with thenatural numerals. For example, any even composite natural numeral may beexpressed as 2k, in which k is a prior natural numeral such that anassociated finite multi-set has been defined, such as above. The finitemulti-set of 2k, here is formed from a union of the finite multi-setassociated with k and the finite multi-set {0}. Likewise, any oddcomposite numeral, k*L, has associated with it the multi-set that isformed from the union of the multi-sets associated with k and L,respectively, for this embodiment. Thus, for this particular embodiment,although the claimed subject matter is not limited in scope in thisrespect, merger of the index of non-composite natural numeralscorresponds to conventional multiplication.

Returning to FIG. 2, the set associated with the natural numeral 4 is aset that includes three zeros {0,0,0}—the merger of the set associatedwith 2 with the set {0}, as explained above. Likewise, the setassociated with the natural numeral 8 comprises four zeros {0,0,0,0} andso forth. Furthermore, the set associated with 6 is the merger of theset associated with 3, here {1}, with the set that contains one zero{0}.

Continuing to column three of FIG. 2, its' elements comprise the finitemulti-sets of column two. Column three, here, is constructed by applyingthe rules described above to column two. Likewise, the rules applied tocolumn three construct column four, etc. It is also noted that analternate way to move from one column to another column is to substitutethe natural numerals in the column with their associated set from columntwo. Thus, for example, for natural numeral 7, its element in the thirdcolumn may be constructed from the second column by replacing 3, whichis along the same row, with its associated set {1}. Likewise to movefrom column three to column four along the same row, the natural numeral1 in {1} is replaced with its associated set {0} to produce {{0}} and soforth.

Although the claimed subject matter is not limited in scope in thisrespect, this ability to move between columns and, hence, betweenmulti-sets, may provide advantages in certain environments. It allowsthe potential for inter-related file structures. For example, in somesituations, it may be desirable to run multiple tasks using related filestructures. Computer science has many examples where this may occur,such as with databases, computer programs, look ahead operations, etc.Thus, an embodiment in which the file structures employs an approachsimilar to that just described, for example, provides the ability tomove between files created as part of performing such disparate butrelated tasks, quickly and easily. Depending upon the particularsituation, it may simply comprise moving between columns in a filestructure, such as the file structure embodiment above, for example.

The previously described approach for constructing multi-sets for thenatural numerals, here column two, provides a technique for associatinga natural numeral, here column one, with a finite multi-set in which theelements of the set are the natural numerals. It may be convenient toview a finite multi-set as a string for the purposes of implementationon a computing device, such as a computer, for example. As will beexplained in more detail hereinafter, this association may be usefullyemployed in a variety of contexts, such as, for this embodiment, toconstruct a file hierarchy, for example.

In this particular embodiment in which a table, such as that illustratedin FIG. 2, is employed to create a file hierarchy, file names areemployed that comprise multi-sets. In this particular embodiment, themulti-set is related to the number of columns to be constructed. In thiscontext, the term “multi-set depth” indicates that the file names aretaken from a particular column such as if the columns are constructed aspreviously described, up to a particular column. For example, if themulti-set depth is 6, ultimately, for this embodiment, it will bedesirable to construct the seventh column for this particularembodiment, as explained in more detail hereinafter. As may be observed,for this embodiment, column two in FIG. 2 has a multi-set depth of 1.For this example, therefore, it is desirable to construct columnscorresponding to depths of 6, 5, 4, 3, 2 and 1, and also to construct aK+1 or seventh column that corresponds to the multi-set depth of K, hereK being 6 to correspond to a multi-set depth of 6. By constructing acolumn of multi-set depth 6, here column 7, a column has been created inwhich the file names employed in this column is related to the filenames in the other prior columns along a corresponding row, for example,although, of course, the claimed subject matter is not limited in scopein this respect.

By this construction, the multi-set file names desired to be employedmay be related to natural numerals by moving between the K+1 column andthe first column. This, therefore, permits movement between hierarchicalfile names in a process that may be implemented, for example, by, forthis embodiment, moving between columns and up and down rows. Such anapproach, thus, may be more efficient than the approach previouslydescribed in connection with the Unix-based operating system at least inpart because it permits direct movement between two files rather thantracing branches of a tree, as previously described. It is likewisenoted that the multi-set depth, or the column from which a particularfile name was selected, may be determined “on the fly,” in thisembodiment, by the number of sets of parentheses in the file name. Ofcourse, the claimed subject matter is not limited to employingparentheses and, thus, other techniques to make such a determination onthe fly are also possible and included within the scope of the claimedsubject matter. However, as noted, for a particular file name, its rowand column location may be determined from the name so that theoperations described above may be quickly and easily implemented if sucha file structure or a similar file structure is employed.

Of course, techniques for performing table look-ups are well-known andwell-understood. Thus, this will not be discussed in detail here.However, it shall be appreciated that any and all of the previouslydescribed and/or later described processing, operations, conversions,transformations, manipulations, etc. of strings, trees, numerals, data,etc. may be performed on one or more computing platforms or similarcomputing devices, such as those that may include a memory to store atable as just described, although, the claimed subject matter is notnecessarily limited in scope to this particular approach. Thus, forexample, a hierarchy of data, such as a tree as previously described,for example, may be formed. Likewise, operations and/or manipulations,as described, may be performed; however, operations and/or manipulationsin addition to those described or instead of those described may also beapplied. It is intended that the claimed subject matter cover all suchembodiments.

For this particular embodiment, although the claimed subject matter isnot limited in scope in this respect, the files may be arranged in anaming hierarchy in which the hierarchy is constructed by recursivelyforming multi-sets of a set of elements. In this particular embodiment,for example, the set of elements may comprise the finite multi-set ofnatural numerals, although, again the claimed subject matter is notlimited in scope in this respect. Recursively forming of multi-sets may,therefore, create a naming hierarchy for files, such as just describedfor this particular embodiment. In this context, recursively formingmulti-sets may also be referred to as compound multi-sets.

FIG. 3 illustrates another embodiment of a table in accordance with theclaimed subject matter. As is apparent from inspection, the first twocolumns of the table of FIG. 2, correspond to the first two columns ofthe table of FIG. 3, although, again, the claimed subject matter is notlimited in scope in this respect. Column one of FIG. 3, therefore, alsocontains natural numerals in ascending order and column two of FIG. 3also contains corresponding finite multi-sets, as previously described.Column three of FIG. 3, however, for this embodiment, contains numeralpairs, as shall be described in more detail hereinafter.

In this particular embodiment, natural numerals of column one arerelated to numeral pairs of column three, as described in more detailbelow. One numeral of the numeral pair contains the index numeral for anassociated non-composite numeral, such as previously described inconnection with the Kleene enumeration. For example, as previouslydescribed, for the natural numeral 3, the index associated is 1.Likewise, for the natural numeral 5, the index associated is 2. For thenatural numeral 7, the index associated is 3, etc.

The other numeral of the numeral pair, for this embodiment, comprisesthe numeral that produces the natural numeral of column one whenmultiplied by the non-composite numeral corresponding to the index ofthe Kleene enumeration. Again, using the natural numeral 3 of column oneas an example, one numeral of the numeral pair is 1, as previouslydescribed. Thus, the other numeral of the numeral pair is also 1. Moreparticularly, 1 times the non-composite numeral, for this example, 3,produces the natural numeral, again in this example 3. A more helpfulmay be the natural numeral 6 of column one. Here, the index may beeither a 1 or a 0, as explained in more detail hereinafter. If the indexis 1, the other natural numeral of the numeral pair is 2, that is, 3(corresponding to 1) times 2 is 6. If, instead, the index is 0, theother numeral is 3 since 2 (corresponding to 0) times 3 is also 6.

As may now be appreciated, column three, in this embodiment, provides amechanism to identify those natural numerals of column one that arenon-composite. In particular, the first numeral is the index describedpreviously, and the second numeral is the numeral that produces thenatural numeral of column one, if multiplied by the non-compositenumeral corresponding to the index. Thus, for this embodiment, numeralpairs in which the second numeral is 1 correspond to those naturalnumerals in column one that are non-composite. It is, again, appreciatedthat the claimed subject matter is not limited in scope to thisparticular embodiment. For example, the numerals may not be arranged incolumns and/or the numeral pair may be reversed. Likewise, instead of anumeral pair in one column, perhaps two columns, each having a numeralmay be employed. In the latter approach, for example, finding a onenumeral in the column will identify the non-composite numerals. Manydifferent approaches and/or embodiments may be employed; however, it isintended that they all be included within the scope of the claimedsubject matter. For this embodiment, the numeral pairs in column threecomprise a first operation in a factorization of the natural numeral incolumn one. Hence, for non-composite numerals, the identified numeralwill be 1, whereas for other natural numerals that are notnon-composite, this particular numeral will be something other than 1.

Column three, in this particular embodiment, also provides a mechanismthrough which it is possible to represent a natural numeral using fewerbits than an associated conventional binary representation. Consider,for example, the natural numeral 9. To represent 9 in a binaryrepresentation employs four bits, as is well-known. However, the columnthree representation of the natural numeral 9 is the numeral pair [1,3],in this particular embodiment. To represent the natural numeral 1employs one bit. To represent the natural numeral 3 employs two bits.Therefore, by substituting the pair [1,3] in memory for 9, for example,three bits may be employed to represent the natural numeral 9, whereasconventionally it takes four bits. As should now be apparent, it ispossible to produce the natural numeral of column one with thecorresponding numeral pair of column three and vice-versa. For example,using again the natural numeral 9, its corresponding numeral pair forthis embodiment is [1,3]. Here, 1 is the index for natural numeral 3, asindicated previously. Thus, 3 (corresponding to one) times 3 provides 9,in this example.

It is noted that, for this embodiment, it is not the case that employingthe numeral pairs of column three always save a bit. However, for allcases, the binary representation associated with the third column,again, for this embodiment, will be less than or equal to the number ofbits conventionally employed to represent the particular naturalnumeral. For example, consider natural numeral 8 in column one. Thecolumn three representation is [0, 4]. Eight is conventionallyrepresented using four bits. Zero is represented using one bit. Four isrepresented using three bits. Therefore, both representations employ thesame number of bits in this example, here, four. Thus, using thisapproach ensures that the same or fewer bits than is conventionally thecase will represent a particular natural numeral p Another feature ofthis particular embodiment, illustrated in FIG. 3, is the association ofa unique finite multi-set with the natural numeral of column one and/orthe numeral pair of column three. Consider, as previously discussed,producing the associated finite multi-set from the particular naturalnumeral. For this particular embodiment, it should also be possible toproduce this finite multi-set from column three. Consider, for example,the case of natural numeral 105. The associated finite multi-set is{1,2,3}. Associated numerical pairs include: [1,35]; [2,21]; and [3,15].Here, the first numeral of the numeral pair is an element of the finitemulti-set for this particular embodiment. The second numeral for thenumeral pair is the product of the non-composite numerals associatedwith the remaining elements of the multi-set. For example, thenon-composite numeral associated with 2 is 5 and the non-compositenumeral associated with 3 is 7. Thus, the product of 5 and 7 is 35.

Thus, for this particular embodiment, the table of FIG. 3 provides, fora particular natural numeral, the finite multi-sets of the naturalnumeral and the numeral pair of the natural numeral. Likewise, thenumeral pair of this embodiment is analogous to the Kleene enumerationextended to include composite numerals. In particular, using thefundamental principle of arithmetic, one may always break any naturalnumeral into a unique set of non-composite numerals. Table three makesuse of this fundamental precept by replacing the non-composites withtheir associated indexes. Furthermore, as previously demonstrated, forany natural numeral, in this embodiment, there is an association betweenthese indexes and the finite multi-sets of column two.

Another feature of this particular embodiment includes employing numeralpairs to perform multiplication. For example, natural numeral 3 isrepresented by the pair [1,1]. Likewise, natural numeral 5 isrepresented by pair [2,1]. In this particular embodiment, a general rulefor combining numeral pairs is as follows:[a,b] combined with [c,d]=[a,b*(Q(c)*d)]  [1]

-   -   in which “*” represents multiplication and where the function        Q(c) is a function that provides the non-composite numeral        associated with the index c, as for the Kleene enumeration.        Thus, in this example, merging the pairs produces the ordered        pair [1,5]. This numeral pair corresponds to the natural numeral        15, which is a product of 3 times 5. Furthermore, since, as        previously demonstrated, column three provides a mechanism to        represent the natural numeral of column one with the same or        fewer bits, this also provides a mechanism to perform        multiplication with the same of fewer bits than conventionally        employed.

For this embodiment, the table of FIG. 3 provides a mechanism tomanipulate finite multi-sets, or employing the language of computerscience, to manipulate strings, using a table-driven mechanism. Forexample, multiplication, such as 3 times 5, combines the finitemulti-setss corresponding to those natural numerals. This may beaccomplished, for this embodiment, therefore, either by multiplying thenatural numerals of column one or by using the merger operation ofcolumn three, as described above. Thus, by a table look up mechanism,strings may be combined, such as in the manner described above for thisparticular embodiment.

Referring now to FIG. 4, a schematic diagram illustrates an embodimentof a binary tree file hierarchy “rooted” upon a finite multi-sets. Asshall be explained in more detail hereinafter, this particularembodiment of a file hierarchy may be constructed based at least in partupon the table of FIG. 3. Consider, for example, the natural numeral 12of column one. In column two, this corresponds to the finite multi-sets{0,0,1}. Using the table of FIG. 3, for this embodiment, associated withthis finite multi-sets, and thus, also associated with natural numeral12, are two unique non-composite numerals. It is noted that there aremany different unique pairs of non-composite numerals that may beassociated with natural numeral 12. This is merely one embodiment andthe claimed subject matter is not limited in scope to this particularembodiment. However, one non-composite numeral comprises Q(2x), or here,Q(2*12), that is, Q(24). Another non-composite numeral comprisesQ(2x−1), here, Q(23). As demonstrated by the binary tree file hierarchyembodiment illustrated in FIG. 4, this technique provides a filehierarchy of named file locations that is well-known to be useful forsearching and/or sorting. See, for example, The Art of ComputerProgramming, Vol. 2: Searching and Sorting, by Donald K. Knuth, 3^(rd)Edition, 1997, available from Addison Wesley Longman. Thus, the table ofFIG. 3 provides an infinite number of binary trees, each binary treeassociated with a finite multi-set associated with two or more naturalnumerals larger than one.

The table of FIG. 3 also provides another approach to moving betweennamed files in a file structure or file hierarchy. For this particularembodiment, this is accomplished by associating at least some namedfiles in the file structure with products of non-composite numerals, andassociating other named files in the file structure with non-compositenumerals. As shall be explained in more detail hereinafter, in such anembodiment, moving between named files in a file hierarchy may beaccomplished via a mathematical operation, such as by dividing and/ormultiplying with non-composite numerals.

As previously described and illustrated by the tree of FIG. 1, in aUnix-based operating system, it may be time consuming in a relativesense to move between files in a file structure, using serial access.FIG. 5 illustrates an embodiment of an alternate file hierarchy in-whichmoving between named files may accomplished through mathematicalmanipulation, thus, potentially saving time. In FIG. 5, the nodesrepresent named files that have particular representations in memory.For this example, let file 510 have associated with it the non-compositenumeral Q(2x) This is analogous or equivalent to root node 110 ofFIG. 1. Emanating from file 510 are thus a potentially infinite numberof named files, such as, for example, files 520 and 530. In thisparticular embodiment, these named files are located serially in memorybased at least in part on a product of two non-composite numerals. Thus,for example, assume it is desirable to move between file 510 and file520. This may be accomplished simply by dividing by the non-compositenumeral Q(2y₁−1) and then multiplying by the non-composite numeralQ(2u_(k)−1). Thus, moving between these named file locations reduces toa mathematical calculation. It is noted that this is simply oneparticular embodiment of this approach and many other embodiments may beemployed. For example, a non-composite numeral other than Q(2x) may beemployed as the core and other named files associated with the core maybe located in a manner other than as illustrated in FIG. 5. Thus, it isintended to cover all such embodiments regardless of the particularnon-composite numerals employed to name and locate such files.

Furthermore, it will, of course, be appreciated that, for convenience,in a particular embodiment, a translation table between characters andnon-composite numerals may also be employed. Thus, in a typicaloperation, a first file may have a named location represented as asequence of characters and a table look-up may be employed to obtain itsdesignation or named location in terms of non-composite numerals orproducts of non-composite numerals, such as an embodiment like the onepreviously described. Mathematical operations may then be applied to thenon-composite numerals or the products of non-composite numerals, insuch an embodiment, to locate the named file it is desired to locate andthen a translation table may be employed to provide the charactersassociated with the resulting non-composite numeral or product ofnon-composite numerals. In this particular embodiment, therefore, theessence of manipulating files or moving between files in a filehierarchy is reduced to division of non-composite numerals and/orproducts of non-composite numerals. In general, therefore, a method ofconstructing a file structure may comprise associating at least somefiles with non-composite numerals. It is, therefore, intended to coverall embodiments of such a file hierarchy.

It will, of course, be understood that, although particular embodimentshave just been described, the claimed subject matter is not limited inscope to a particular embodiment or implementation. For example, oneembodiment may be in hardware, such as implemented to operate on adevice or combination of devices, for example, whereas anotherembodiment may be in software. Likewise, an embodiment may beimplemented in firmware, or as any combination of hardware, software,and/or firmware, for example. Likewise, although the claimed subjectmatter is not limited in scope in this respect, one embodiment maycomprise one or more articles, such as a storage medium or storagemedia. This storage media, such as, one or more CD-ROMs and/or disks,for example, may have stored thereon instructions, that when executed bya system, such as a computer system, computing platform, or othersystem, for example, may result in an embodiment of a method inaccordance with the claimed subject matter being executed, such as oneof the embodiments previously described, for example. As one potentialexample, a computing platform may include one or more processing unitsor processors, one or more input/output devices, such as a display, akeyboard and/or a mouse, and/or one or more memories, such as staticrandom access memory, dynamic random access memory, flash memory, and/ora hard drive. For example, a display may be employed to display one ormore queries, such as those that may be interrelated, and or one or moretree expressions, although, again, the claimed subject matter is notlimited in scope to this example.

In the preceding description, various aspects of the claimed subjectmatter have been described. For purposes of explanation, specificnumbers, systems and/or configurations were set forth to provide athorough understanding of the claimed subject matter. However, it shouldbe apparent to one skilled in the art having the benefit of thisdisclosure that the claimed subject matter may be practiced without thespecific details. In other instances, well-known features were omittedand/or simplified so as not to obscure the claimed subject matter. Whilecertain features have been illustrated and/or described herein, manymodifications, substitutions, changes and/or equivalents will now occurto those skilled in the art. It is, therefore, to be understood that theappended claims are intended to cover all such modifications and/orchanges as fall within the true spirit of the claimed subject matter.

1. An article comprising: a storage medium; said storage medium havingstored thereon a file location naming hierarchy; wherein said hierarchycomprises at least a multi-set (MS) of a set of elements.
 2. The articleof claim 1, wherein said MS of a set of elements comprises a MS of a MSof said set of elements.
 3. The article of claim 1, wherein said set ofelements comprise natural numerals.
 4. The article of claim 1, whereinsaid hierarchy further comprises compound multi-sets of said set ofelements.
 5. A method comprising: forming a file location naminghierarchy; wherein said hierarchy comprises at least a multi-set (MS) ofa set of elements.
 6. The method of claim 5, wherein said MS of a set ofelements comprises a MS of a MS of said set of elements.
 7. The methodof claim 5, wherein said set of elements comprise natural numerals. 8.The method of claim 5, wherein said hierarchy further comprises compoundmulti-sets of said set of elements.
 9. A method of arranging files in afile location naming hierarchy comprising: forming a multi-set from aset of elements; forming at least one more multi-set from saidmulti-set; wherein said hierarchy comprises the multi-sets.
 10. Themethod of claim 9, wherein said forming at least one more multi-setcomprises recursively forming at least one compound multi-set.
 11. Themethod of claim 10, wherein said recursively forming at least onecompound multi-set comprises recursively forming a plurality of compoundmulti-sets.
 12. An article comprising: a storage medium having storedthereon instructions that, when executed, result in execution of amethod of arranging files in a file location naming hierarchy asfollows: forming a multi-set from a set of elements; forming at leastone more multi-set from said multi-set; wherein said hierarchy comprisesthe multi-sets.
 13. The article of claim 13, wherein said instructionswhen executed further result in: said forming at least one moremulti-set comprising recursively forming at least one compoundmulti-set.
 14. The article of claim 13, wherein said instructions, whenexecuted, further result in: said recursively forming at least onecompound multi-set comprising recursively forming a plurality ofcompound multi-sets.
 15. A method of associating a natural numeral witha multi-set comprising: for the numeral 2, assigning a unique multi-set;and for even composite numerals 2K, assigning the merger of themulti-set for 2 with the multi-set for K.
 16. The method of claim 15,and further comprising: for the numerals 0 and 1, assigning additionalunique multi-sets; and for odd non-composite numerals, assigning theassociated Kleene enumeration index.
 17. The method of claim 16, andfurther comprising: for composite numerals K*L, assigning the merger ofthe multi-set for K with the multi-set for L.
 18. The method of claim16, wherein, for the numerals 0 and 1, assigning additional uniquemulti-sets comprises assigning the empty set and singleton zero,respectively.
 19. The method of claim 15, wherein, for the numeral 2,assigning a unique multi-set comprises assigning doubleton zero.
 20. Anarticle comprising: a storage medium having stored thereon instructionsthat, when executed, result in a method of associating a natural numeralwith a multi-set as follows: for the numeral 2, assigning a uniquemulti-set; and for even composite numerals 2K, assigning the merger ofthe multi-set for 2 with the multi-set for K.
 21. The article of claim20, wherein said instructions, when executed, further result in: for thenumerals 0 and 1, assigning additional unique multi-sets; and for oddnon-composite numerals, assigning the associated Kleene enumerationindex.
 22. The article of claim 21, wherein said instructions, whenexecuted, further result in: for composite numerals, K*L, assigning themerger of the multi-set for K with the multi-set for L.
 23. The articleof claim 21, wherein said instructions, when executed, further resultin: for the numerals 0 and 1, assigning additional unique multi-setscomprising assigning the empty set and singleton zero, respectively. 24.The article of claim 20, wherein said instructions, when executed,further result in: for the numeral 2, assigning a unique multi-setcomprising assigning doubleton zero.
 25. A method of associating anatural numeral with a numeral pair comprising: forming said numeralpair so that it includes a Kleene enumeration index natural numeralcorresponding to a non-composite numeral and another numeral thatproduces said natural numeral when multiplied with said non-compositenumeral.
 26. The method of claim 25, wherein said natural numeral thatcomprises a non-composite numeral is designated by said another numeralof said numeral pair comprising the numeral
 1. 27. The method of claim25, and further comprising: forming a finite multi-set from said numeralpair.
 28. The method of claim 27, and further comprising: manipulatingsaid finite multi-set.
 29. The method of claim 28, and furthercomprising: forming another finite multi-set from another numeral pair;wherein said manipulating said finite multi-set comprises: combining thefinite multi-sets.
 30. The method of claim 29, wherein said combiningthe finite multi-sets comprises: multiplying the associated naturalnumerals.
 31. The method of claim 29, wherein said combining the finitemulti-sets comprises: merging the associated numeral pairs.
 32. Themethod of claim 25, and further comprising: associating another naturalnumeral with another numeral pair; and multiplying the natural numeralsby merging the associated numeral pairs.
 33. The method of claim 32,wherein said merging numeral pairs comprises: combining [a,b] with [c,d]as follows: [a, b*Q(c)″*d], in which “*” represents multiplication andwhere the function Q(c) is a function that provides the non-compositenumeral associated with the index c, as for the Kleene enumeration. 34.The method of claim 32, wherein said numeral pairs are represented asbinary numerals, and said multiplying said natural numerals includesusing the same or fewer bits than associated with binary representationsof said natural numerals by merging the binary numeral pairs.
 35. Themethod of claim 25, wherein said natural numeral is represented by abinary numeral representation of said numeral pair.
 36. An articlecomprising: a storage medium having stored thereon instructions that,when executed, result in execution of a method of associating a naturalnumeral with a numeral pair as follows: forming said numeral pair sothat it includes a Kleene enumeration index natural numeralcorresponding to a non-composite numeral and another numeral thatproduces said natural numeral when multiplied with said non-compositenumeral.
 37. The article of claim 36, wherein said instructions, whenexecuted, further result in: said natural numeral that comprises anon-composite numeral being designated by said another numeral of saidnumeral pair comprising a numeral
 1. 38. The article of claim 36,wherein said instructions, when executed, further result in: forming afinite multi-set from said numeral pair.
 39. The article of claim 38,wherein said instructions, when executed, further result in:manipulating said finite multi-set.
 40. The article of claim 39, whereinsaid instructions, when executed, further result in: forming anotherfinite multi-set from another numeral pair; wherein manipulating saidfinite multi-set comprises: combining the finite multi-sets.
 41. Thearticle of claim 40, wherein said instructions, when executed, furtherresult in: said combining the finite multi-sets comprising multiplyingthe associated natural numerals.
 42. The article of claim 40, whereinsaid instructions, when executed, further result in: said combining thefinite multi-sets comprising merging the associated numeral pairs. 43.The article of claim 36, wherein said instructions, when executed,further result in: associating another natural numeral with anothernumeral pair; and multiplying the natural numerals by merging theassociated numeral pairs.
 44. The article of claim 43, wherein saidinstructions, when executed, further result in: said merging theassociated numeral pairs comprising combining [a,b] with [c,d] asfollows: [a, b*Q(c)″*d], in which “*” represents multiplication andwhere the function Q(c) is a function that provides the non-compositenumeral associated with the index c, as for the Kleene enumeration. 45.The article of claim 43, wherein said instructions, when executed,further result in: said numeral pairs being represented as binarynumerals: and said multiplying said natural numerals including using thesame or fewer bits than associated with binary representations of saidnatural numerals by merging the binary numeral pairs.
 46. The article ofclaim 36, wherein said instructions, when executed, further result in:said natural numeral being represented by a binary numeralrepresentation of said numeral pair.
 47. An article comprising: astorage medium having stored thereon a table; wherein said tableuniquely associates a plurality of natural numerals with a plurality ofnumeral pairs, wherein at least one of the numerals for each of saidpairs represents an index for the Kleene enumeration.
 48. The article ofclaim 47, wherein said table further uniquely associates a plurality offinite multi-sets with said natural numerals and said numeral pairs. 49.The article of claim 48, wherein, for the numerals 0, 1, and 2, theassociated multi-sets comprise the empty set, singleton zero anddoubleton zero, respectively.
 50. The article of claim 49, wherein, forcomposite numerals 2K, the associated multi-set comprises a merger ofsingleton zero with the multi-set associated with K.
 51. The article ofclaim 50, wherein, for composite numerals K*L, the associated multi-setcomprises a merger of the multi-set associated with K with the multi-setassociated with L.
 52. A method comprising: uniquely associating aplurality of natural numerals with a plurality of numeral pairs, whereinat least one of the numerals for each of said pairs represents an indexfor the Kleene enumeration.
 53. The method of claim 52, wherein uniquelyassociating further comprises uniquely associating a plurality of finitemulti-sets with said natural numerals and said numeral pairs.
 54. Themethod of claim 53, wherein uniquely associating, for the numerals 0, 1,and 2, comprises associating the multi-sets empty set, singleton zeroand doubleton zero, respectively.
 55. The method of claim 54, whereinuniquely associating, for composite numerals 2K, comprises associatingthe multi-set comprising a merger of singleton zero with the multi-setassociated with K.
 56. The method of claim 55, wherein uniquelyassociating, for composite numerals K*L, comprises associating themulti-set comprising a merger of the multi-set associated with K withthe multi-set associated with L.
 57. A method of forming a binary treefile location naming hierarchy from a finite multi-set comprising:associating said finite multi-set with a natural numeral; andassociating a first non-composite numeral and a second non-compositenumeral with said natural numeral, said non-composite numerals and saidfinite multi-set forming at least a portion of said hierarchy.
 58. Themethod of claim 57, wherein said finite multi-set represents a namedlocation for a file in said hierarchy, and said first and secondnon-composite numerals represent additional named locations for relatedfiles in said hierarchy.
 59. The method of claim 57, and furthercomprising associating a plurality of binary trees with finitemulti-sets, each binary tree associated with a finite multi-set furtherassociated with two or more natural numerals larger than one.
 60. Anarticle comprising: a storage medium having stored thereon instructionsthat, when executed, result in: a method of forming a binary tree filelocation naming hierarchy from a finite multi-set as follows:associating said finite multi-set with a natural numeral; andassociating a first non-composite numeral and a second non-compositenumeral with said natural numeral, said non-composite numerals and saidfinite multi-set forming at least a portion of said hierarchy.
 61. Thearticle of claim 60, wherein said instructions, when executed, furtherresult in: said finite multi-set representing a named location for afile in said hierarchy, and said first and second non-composite numeralsrepresenting additional named locations for related files in saidhierarchy.
 62. The article of claim 60, wherein said instructions, whenexecuted, further result in: associating a plurality of binary treeswith finite multi-sets, each binary tree associated with a finitemulti-set further associated with two or more natural numerals largerthan one.
 63. A method comprising: associating at least some files of afile location naming hierarchy with non-composite numerals.
 64. Themethod of claim 63, wherein said associating comprises associating saidat least some files in said file location naming hierarchy withnon-composite numerals and/or products of non-composite numerals; andfurther comprising: moving between said at least some files includingdividing and/or multiplying said non-composite numerals and/orassociated products with non-composite numerals.
 65. The method of claim64 wherein said associating comprises associating other files other thansaid at least some files with non-composite numerals.
 66. The method ofclaim 65, wherein said moving between said at least some filescomprising dividing and/or multiplying with non-composite numerals. 67.The method of claim 64, wherein said associating comprises associatingsaid at least some files with non-composite numerals; and said movingbetween said at least some files comprising multiplying and/or dividingby a non-composite numeral.
 68. An article comprising: a storage mediumhaving stored thereon instructions that, when executed results inexecution of a method as follows: associating at least some files of afile location naming hierarchy with non-composite numerals.
 69. Thearticle of claim 68, wherein said instructions when executed furtherresult in: said associating comprising associating said at least somefiles in said file location naming hierarchy with non-composite numeralsand/or products of non-composite numerals; and further result in: movingbetween said at least some files including dividing and/or multiplyingsaid non-composite numerals and/or associated products withnon-composite numerals.
 70. The article of claim 69, wherein saidinstructions when executed further result in: said associatingcomprising associating other files other than said at least some fileswith non-composite numerals.
 71. The article of claim 70, wherein saidinstructions when executed further result in: said moving between saidat least some files comprising dividing and/or multiplying withnon-composite numerals.
 72. The article of claim 69, wherein saidinstructions when executed further result in: said associatingcomprising associating said at least some files with non-compositenumerals; and said moving between said at least some files comprisingmultiplying and/or dividing by a non-composite numeral.
 73. An apparatuscomprising: a computing platform; said computing platform having storedthereon a file location naming hierarchy; wherein said hierarchycomprises at least a multi-set (MS) of a set of elements.
 74. Theapparatus of claim 73, wherein said MS of a set of elements comprises aMS of a MS of said set of elements.
 75. The apparatus of claim 73,wherein said set of elements comprise natural numerals.
 76. Theapparatus of claim 73, wherein said hierarchy further comprises compoundmulti-sets of said set of elements.
 77. An article comprising: a filehaving a named file location; said named location having been formed atleast by: forming a file location naming hierarchy; wherein saidhierarchy comprises at least a multi-set (MS) of a set of elements. 78.The article of claim 77, wherein said file having a named locationhaving further been formed at least by: said MS of a set of elementscomprising a MS of a MS of said set of elements.
 79. The article ofclaim 77, wherein said file having a named location having further beenformed at least by: said set of elements comprising natural numerals.80. The article of claim 77, wherein said file having a named locationhaving further been formed at least by: said hierarchy comprisingcompound multi-sets of said set of elements.
 81. An article comprising:a file having a named location having been arranged in a file locationnaming hierarchy having been formed at least by: forming a multi-setfrom a set of elements; forming at least one more multi-set from saidmulti-set; said hierarchy comprising the multi-sets.
 82. The article ofclaim 81, wherein said named location having further been formed atleast by forming at least one more multi-set comprising recursivelyforming at least one compound multi-set.
 83. The article of claim 82,wherein said named location having further been formed at least by saidrecursively forming at least one compound comprising recursively forminga plurality of compound multi-sets.
 84. An apparatus comprising: acomputing platform; said computing platform adapted to form a multi-setfrom a set of elements; adapted to form at least one more multi-set fromsaid multi-set; and adapted so that said hierarchy comprises themulti-sets.
 85. The apparatus of claim 84, wherein said computingplatform is adapted to form at least one more multi-set by recursivelyforming at least one compound multi-set.
 86. The apparatus of claim 85,wherein said computing platform is adapted to recursively forming atleast one compound multi-set by recursively forming a plurality ofcompound multi-sets.
 87. An article comprising: a file having a namedlocation; said named location being formed from a multi-set associatedwith a natural numeral at least by: for 2, assigning a unique multi-set;and for even composite numerals, 2K, assigning the merger of themulti-set for 2 with the multi-set for K.
 88. The article of claim 87,said named location being formed from a multi-set further associatedwith a natural numeral at least by: for 0 and 1, assigning additionalunique multi-sets; and for odd non-composite numerals, assigning theassociated Kleene enumeration index.
 89. The article of claim 88, saidnamed location being formed from a multi-set further associated with anatural numeral at least by: for composite numerals, K×L, assigning themerger of the multi-set for K with the multi-set for L.
 90. The articleof claim 88, said named location being formed from a multi-set furtherassociated with a natural numeral at least by: for 0 and 1, assigningunique multi-sets comprises assigning the empty and singleton zero,respectively.
 91. The article of claim 87, said named location beingformed from a multi-set further associated with a natural numeral atleast by: for 2, assigning a unique multi-set comprises assigningdoubleton zero.
 92. An apparatus comprising: a computing platform; saidcomputing platform being adapted to associate a natural numeral with amulti-set as follows: for 2, assigning a unique multi-set; and for evencomposite numerals, 2K, assigning the merger of the multi-set for 2 withthe multi-set for K.
 93. The apparatus of claim 92, wherein saidcomputing platform is further adapted to: for 0 and 1, assign additionalunique multi-sets; and for odd non-composite numerals, assign theassociated Kleene enumeration index.
 94. The apparatus of claim 93,wherein said computing platform is further adapted to: for compositenumerals, K×L, assign the merger of the multi-set for K with themulti-set for L.
 95. The apparatus of claim 93, wherein said computingplatform is further adapted to: for 0 and 1, assign unique multi-setscomprising the empty set and singleton zero, respectively.
 96. Theapparatus of claim 92, wherein said computing platform is furtheradapted to: for 2, assign a unique multi-set comprising doubleton zero.97. An article comprising: a file having a named location; said namedlocation being formed from a numerical pair associated with a naturalnumeral at least by: forming said numeral pair so that it includes anindex natural numeral corresponding to a non-composite numeral andanother numeral that produces said natural numeral when multiplied withsaid non-composite numeral.
 98. The article of claim 97, wherein saidnamed location being formed from a numerical pair further associatedwith a natural numeral at least by: said natural numeral that comprisesa non-composite numeral being designated by said another numeral of saidnumeral pair comprising a one numeral.
 99. The article of claim 97,wherein said named location being formed from a numerical pair furtherassociated with a natural numeral at least by: forming a finitemulti-set from said numeral pair.
 100. The article of claim 99, saidnamed location being formed from a numerical pair further associatedwith a natural numeral at least by: manipulating said finite multi-set.101. The article of claim 100, said named location being formed from anumerical pair further associated with a natural numeral at least by:forming another finite multi-set from another numeral pair; whereinmanipulating said finite multi-set comprises: combining said finitemulti-sets.
 102. The article of claim 101, wherein said named locationbeing formed from a numerical pair associated with a natural numeral atleast by: said combining said finite multi-sets comprising multiplyingthe associated natural numerals.
 103. The article of claim 101, whereinsaid named location being formed from a numerical pair furtherassociated with a natural numeral at least by: said combining saidfinite multi-sets comprising merging the associated numeral pairs. 104.The article of claim 97, said named location being formed from anumerical pair further associated with a natural numeral at least by:associating another natural numeral with another numeral pair; andmultiplying natural numerals by merging associated numeral pairs. 105.The article of claim 104, wherein said named location being formed froma numerical pair further associated with a natural numeral at least by:said merging numeral pairs comprising combining [a,b] with [c,d] asfollows: [a, b*Q(c)″*d], in which “*” represents multiplication andwhere the function Q(c) is a function that provides the non-compositenumeral associated with the index c, as for the Kleene enumeration. 106.The article of claim 104, wherein said named location being formed froma numerical pair further associated with a natural numeral at least by:said numeral pairs being represented as binary numerals, and multiplyingsaid natural numerals using the same or fewer bits than associated withbinary representations of said natural numerals by merging the binarynumeral pairs.
 107. The article of claim 97, wherein said named locationbeing formed from a numerical pair further associated with a naturalnumeral at least by: said natural numeral being represented by a binarynumeral representation of said numeral pair.
 108. An apparatuscomprising: a computing platform; said computing platform being adaptedto associate a natural numeral with a numeral pair as follows: formingsaid numeral pair so that it includes an index natural numeralcorresponding to a non-composite numeral and another numeral thatproduces said natural numeral when multiplied with said non-compositenumeral.
 109. The apparatus of claim 108, wherein said computingplatform being further adapted to associate a natural numeral with anumeral pair by said natural numeral that comprises a non-compositenumeral being designated by said another numeral of said numeral paircomprising a one numeral.
 110. The apparatus of claim 108, wherein saidcomputing platform is further adapted to form a finite multi-set fromsaid numeral pair.
 111. The apparatus of claim 110, wherein saidcomputing platform is further adapted to manipulate said finitemulti-set.
 112. The apparatus of claim 11 1, wherein said computingplatform is further adapted to form another finite multi-set fromanother numeral pair; wherein manipulating said finite multi-setcomprises combining said finite multi-sets.
 113. The apparatus of claim112, wherein said computing platform is further adapted to combine saidfinite multi-sets by multiplying the associated natural numerals. 114.The apparatus of claim 112, wherein said computing platform is furtheradapted to combine said finite multi-sets by merging the associatednumeral pairs.
 115. The apparatus of claim 108, wherein said computingplatform is further adapted to associate another natural numeral withanother numeral pair; and to multiply the natural numerals by mergingassociated numeral pairs.
 116. The apparatus of claim 115, wherein saidcomputing platform is further adapted to combine [a,b] with [c,d] asfollows: [a, b*Q(c)″*d], in which “*” represents multiplication andwhere the function Q(c) is a function that provides the non-compositenumeral associated with the index c, as for the Kleene enumeration. 117.The apparatus of claim 115, wherein said computing platform is furtheradapted to represent said numeral pairs as binary numerals: and tomultiply said natural numerals using the same or fewer bits thanassociated with binary representations of said natural numerals bymerging the binary numeral pairs.
 118. The apparatus of claim108,wherein said computing platform being adapted to represent said naturalnumeral by a binary numeral representation of said numeral pair.
 119. Anapparatus comprising: a computing platform; said computing platformadapted to uniquely associate a plurality of natural numerals with aplurality of numeral pairs, wherein at least one of the numerals foreach of said pairs represents an index for the Kleene enumeration. 120.The apparatus of claim 119, wherein said computing platform is furtheradapted to uniquely associate a plurality of finite multi-sets with saidnatural numerals and said numeral pairs.
 121. The apparatus of claim120, wherein said computing platform is further adapted to associate,for the numerals 0, 1, and 2, the multi-sets, the empty set, singletonzero and doubleton zero respectively.
 122. The apparatus of claim 121,wherein computing platform is further adapted to associated, forcomposite numerals, 2K, the multi-set comprising a merger of singletonzero with the multi-set associated with K.
 123. The apparatus of claim122, wherein said computing platform is further adapted to associate,for composite numerals, K*L, the multi-set comprising a merger of themulti-set associated with K with the multi-set associated with L. 124.An article comprising: a file; said file having a named location; saidnamed location being part of a file location naming hierarchy formed atleast by: associating a plurality of natural numerals with a pluralityof numeral pairs, wherein at least one of the numerals for each of saidpairs represents an index for the Kleene enumeration.
 125. The articleof claim 124, wherein said file location naming hierarchy is furtherformed by: uniquely associating a plurality of finite multi-sets withsaid natural numerals and said numeral pairs.
 126. The article of claim125, wherein said file location naming hierarchy is further formed by:associating, for the numerals 0, 1, and 2, the multi-sets the empty set,singleton zero and doubleton zero, respectively.
 127. The article ofclaim 126, wherein said file location naming hierarchy is further formedby: associating, for composite numerals 2K, the multi-set comprising amerger of singleton zero with the multi-set associated with K.
 128. Thearticle of claim127, wherein said file location naming hierarchy isfurther formed by: associating, for composite numerals K*L, themulti-set comprising a merger of the multi-set associated with K withthe multi-set associated with L.
 129. An article comprising: a file;said file having a named location; said named location being part of abinary tree file location naming hierarchy at least formed from a finitemulti-set by: associating said finite multi-set with a natural numeral;and associating a first non-composite numeral and a second non-compositenumeral with said natural numeral, said non-composite numerals and saidfinite multi-set forming at least a portion of said hierarchy.
 130. Thearticle of claim 129, wherein said file location naming hierarchy isfurther formed by: said finite multi-set representing a named locationfor a file in said hierarchy, and said first and second non-compositenumerals representing additional named locations for related files insaid hierarchy.
 131. The article of claim 129, wherein said filelocation naming hierarchy is further formed by: associating a pluralityof binary trees with finite multi-sets, each binary tree associated witha finite multi-set further associated with two or more natural numeralslarger than one.
 132. An apparatus comprising: a computing platform;said computing platform adapted to form a binary tree file locationnaming hierarchy from a finite multi-set as follows: associating saidfinite multi-set with a natural numeral; associating a firstnon-composite numeral and a second non-composite numeral with saidnatural numeral, said non-composite numerals and said finite multi-setforming at least a portion of said hierarchy.
 133. The apparatus ofclaim 132, wherein said computing platform being further adapted torepresent a named location for a file in said hierarchy with said finitemulti-set, and to represent additional named locations for related filesin said hierarchy with said first and second non-composite numerals.134. The apparatus of claim 132, wherein said computing platform isfurther adapted to associate a plurality of number of binary trees withfinite multi-sets, each binary tree associated with a finite multi-setfurther associated with two or more natural numerals larger than one.135. An article comprising: a plurality of files; said files having anamed location in a file location naming hierarchy; said hierarchy beingformed at least by: associating at least some files of said filelocation naming hierarchy with non-composite numerals.
 136. The articleof claim 135, wherein said hierarchy is further formed at least by:associating said at least some files in said file location naminghierarchy with non-composite numerals and/or products of non-compositenumerals; and wherein at least some files in said hierarchy are movedbetween by: dividing and/or multiplying said non-composite numeralsand/or associated products with non-composite numerals.
 137. The articleof claim 136 wherein said hierarchy is further formed at least by: saidassociating comprising associating other files other than said at leastsome files with non-composite numerals.
 138. The article of claim 137,wherein at least some files in said hierarchy are moved between by:dividing and/or multiplying with non-composite numerals.
 139. Thearticle of claim 136, wherein said hierarchy is further formed at leastby: said associating comprising associating said at least some fileswith non-composite numerals; and wherein at least some files in saidhierarchy are moved between by: multiplying and/or dividing by anon-composite numeral.
 140. An apparatus comprising: a computingplatform; said computing platform being adapted to: associate at leastsome files of a file location naming hierarchy with non-compositenumerals.
 141. The apparatus of claim 140, wherein said computingplatform is further adapted to associate said at least some files insaid file location naming hierarchy with non-composite numerals and/orproducts of non-composite numerals; and is further adapted to movebetween said at least some files by dividing and/or multiplying saidnon-composite numerals and/or associated products with non-compositenumerals.
 142. The apparatus of claim 141, said computing platform isfurther adapted to associate other files other than said at least somefiles with non-composite numerals.
 143. The apparatus of claim 142, saidcomputing platform is further adapted to move between said at least somefiles by dividing and/or multiplying with non-composite numerals. 144.The apparatus of claim 141, wherein said computing platform is furtheradapted to associate said at least some files with non-compositenumerals; and further adapted to movie between said at least some filesby multiplying and/or dividing by a non-composite numeral.
 145. Anapparatus comprising: means for forming a file location naminghierarchy; wherein said hierarchy comprises at least a multi-set (MS) ofa set of elements; and means for accessing a file in said hierarchy.146. The apparatus of claim 145, wherein said MS of a set of elementscomprises a MS of a MS of said set of elements.
 147. The apparatus ofclaim 145, wherein said set of elements comprise natural numerals. 148.The apparatus of claim 145, wherein said hierarchy further comprisescompound multi-sets of said set of elements.
 149. An apparatuscomprising: means for arranging files in a file location naminghierarchy comprising: means for forming a multi-set from a set ofelements; and means for forming at least one more multi-set from saidmulti-set; wherein said hierarchy comprises the multi-sets.
 150. Theapparatus of claim 149, wherein said means for forming at least one moremulti-set comprising means for recursively forming at least one compoundmulti-set.
 151. The apparatus of claim 150, wherein said means forrecursively forming at least one compound multi-set comprising means forrecursively forming a plurality of compound multi-sets.
 152. Anapparatus comprising: means for associating a natural numeral with amulti-set comprising: means for assigning a unique multi-set, for thenumeral 2; and means for assigning the merger of the multi-set for 2with the multi-set for K, for even composite numerals 2K.
 153. Theapparatus of claim 152, and further comprising: means for assigningadditional unique multi-sets, for the numerals 0 and 1; and means forassigning the associated Kleene enumeration index, for odd non-compositenumerals.
 154. The apparatus of claim 153, and further comprising: meansfor assigning the merger of the multi-set for K with the multi-set forL, for composite numerals K*L.
 155. The apparatus of claim 153, whereinsaid means for assigning additional unique multi-sets comprising meansfor assigning the empty set and singleton zero, for the numerals 0 and1, respectively.
 156. The apparatus of claim 152, wherein said means forassigning a unique multi-set comprising means for assigning doubletonzero, for the numeral
 2. 157. An apparatus comprising: means forassociating a natural numeral with a numeral pair comprising: means forforming said numeral pair; said means for forming said numeral paircomprising: means for including a Kleene enumeration index naturalnumeral corresponding to a non-composite numeral and means for includinganother numeral that produces said natural numeral when multiplied withsaid non-composite numeral.
 158. The apparatus of claim 157, and furthercomprising: means for designating said natural numeral that comprises anon-composite numeral; said means for designating comprising means forsaid another numeral of said numeral pair including a numeral
 1. 159.The apparatus of claim 157, and further comprising: means for forming afinite multi-set from said numeral pair.
 160. The apparatus of claim159, and further comprising: means for manipulating said finitemulti-set.
 161. The apparatus of claim 160, and further comprising:means for forming another finite multi-set from another numeral pair;wherein said means for manipulating said finite multi-set comprises:means for combining the finite multi-sets.
 162. The apparatus of claim161, wherein said means for combining the finite multi-sets comprising:means for multiplying the associated natural numerals.
 163. Theapparatus of claim 161, wherein said means for combining the finitemulti-sets comprising: means for merging the associated numeral pairs.164. The apparatus of claim 157, and further comprising: means forassociating another natural numeral with another numeral pair; and meansmultiplying the natural numerals comprising means for merging theassociated numeral pairs.
 165. The apparatus of claim 164, wherein saidmeans for merging the associated numeral pairs comprising: means forcombining [a,b] with [c,d]; wherein said means for combining comprises[a, b*Q(c)″*d], in which “*” represents multiplication and where thefunction Q(c) is a function that provides the non-composite numeralassociated with the index c, as for the Kleene enumeration.
 166. Theapparatus of claim 164, and further comprising: means for representingsaid numeral pairs as binary numerals; and wherein said means formultiplying said natural numerals including means for merging the binarynumeral pairs using the same or fewer bits than associated with binaryrepresentations of said natural numerals.
 167. The apparatus of claim157, and further comprising: means for representing said natural numeralwith a binary numeral representation of said numeral pair.
 168. Anapparatus comprising: means for uniquely associating a plurality ofnatural numerals with a plurality of numeral pairs, wherein at least oneof the numerals for each of said pairs represents an index for theKleene enumeration.
 169. The apparatus of claim 168, wherein said meansfor uniquely associating comprises means for uniquely associating aplurality of finite multi-sets with said natural numerals and saidnumeral pairs.
 170. The apparatus of claim 169, and further comprising:means for uniquely associating, for the numerals 0, 1, and 2, themulti-sets comprising the empty set, singleton zero and doubleton zero,respectively.
 171. The apparatus of claim 170, and further comprising:means for uniquely associating, for composite numerals 2K, the multi-setcomprising a merger of singleton zero with the multi-set associated withK.
 172. The apparatus of claim 171, and further comprising: means foruniquely associating, for composite numerals K*L, the multi-setcomprising a merger of the multi-set associated with K with themulti-set associated with L.
 173. An apparatus comprising: means forminga binary tree file location naming hierarchy from a finite multi-setcomprising: means for associating said finite multi-set with a naturalnumeral; and means for associating a first non-composite numeral and asecond non-composite numeral with said natural numeral, saidnon-composite numerals and said finite multi-set forming at least aportion of said hierarchy.
 174. The apparatus of claim 173, and furthercomprising: means for representing a named location for a file in saidhierarchy with said finite multi-set, and means for representingadditional named locations for related files in said hierarchy with saidfirst and second non-composite numerals.
 175. The apparatus of claim173, and further comprising: means for associating a plurality of binarytrees with finite multi-sets, each binary tree associated with a finitemulti-set further associated with two or more natural numerals largerthan one.
 176. An apparatus comprising: means for associating at leastsome files of a file location naming hierarchy with non-compositenumerals.
 177. The apparatus of claim 176, wherein said means forassociating comprising means for associating said at least some files insaid file location naming hierarchy with non-composite numerals and/orproducts of non-composite numerals; and further comprising: means formoving between said at least some files including dividing and/ormultiplying said non-composite numerals and/or associated products withnon-composite numerals.
 178. The apparatus of claim 177, wherein saidmeans for associating comprising means for associating other files otherthan said at least some files with non-composite numerals.
 179. Theapparatus of claim 178, said means for moving between said at least somefiles comprising dividing and/or multiplying with non-compositenumerals.
 180. The apparatus of claim 177, wherein said means forassociating comprising means for associating said at least some fileswith non-composite numerals; and said means for moving between said atleast some files comprising multiplying and/or dividing by anon-composite numeral.